The H\'{e}non-Heiles potential is undoubtedly one of the most simple,
classical and characteristic Hamiltonian systems. The aim of this work is to
reveal the influence of the value of the total orbital energy, which is the
only parameter of the system, on the different families of orbits, by
monitoring how the percentage of chaotic orbits, as well as the percentages of
orbits composing the main regular families evolve when energy varies. In
particular, we conduct a thorough numerical investigation distinguishing
between ordered and chaotic orbits, considering only bounded motion for several
energy levels. The smaller alignment index (SALI) was computed by numerically
integrating the equations of motion as well as the variational equations to
extensive samples of orbits in order to distinguish safely between ordered and
chaotic motion. In addition, a method based on the concept of spectral dynamics
that utilizes the Fourier transform of the time series of each coordinate is
used to identify the various families of regular orbits and also to recognize
the secondary resonances that bifurcate from them. Our exploration takes place
both in the physical $(x,y)$ and the phase $(y,\dot{y})$ space for a better
understanding of the orbital properties of the system. It was found, that for
low energy levels the motion is entirely regular being the box orbits the most
populated family, while as the value of the energy increases chaos and several
resonant families appear. We also observed, that the vast majority of the
resonant orbits belong in fact in bifurcated families of the main 1:1 resonant
family. We have also compared our results with previous similar outcomes
obtained using different chaos indicators.

通过计算混沌动力学指标和谱动力学方法，研究了 Hamiltonian 系统的能量对轨道种类的影响，发现随着能量的增加，不同的谐振家族会分裂成次级的，小于 1:1 的谐振次级，产生混沌运动。